Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=\sqrt{x}, \quad g(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

The composite functions \(f \circ g\) and \(g \circ f\) both equal to \(x^{1/4}\), with the domain being \(x \geq 0\). The graphs of these functions would be identical, hence \(f \circ g = g \circ f\).

Step 1: Compute \(f \circ g\) and \(g \circ f\)

To find \(f(g(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}\). Similarly, \(g(f(x)) = g(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}\). So, \(f \circ g = g \circ f = x^{1/4}\).

Step 2: Determine the domain of \(f \circ g\)

The domain of the function \(f \circ g = x^{1/4}\) is the set of all real numbers \(x\) for which \(x^{1/4}\) is defined. As the fourth root of a number is defined for all non-negative \(x\), the domain of \(f \circ g\) is \(x \geq 0\).

Step 3: Graph \(f \circ g\) and \(g \circ f\)

Using a graphing utility, plot the function \(y = x^{1/4}\) to graph \(f \circ g\) and \(g \circ f\). Here, the x-axis is limited to \(x \geq 0\) as per the domain determined in step 2.

Step 4: Determine if \(f \circ g = g \circ f\)

From step 1, it appears that \(f \circ g\) and \(g \circ f\) produce the same result, \(x^{1/4}\). This proves that \(f \circ g = g \circ f\), as these expressions are equivalent for function composition.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually denoted as \(x\)) that make the function work without any issues.
For example, when dealing with square roots, we want to ensure that the numbers under the root are positive because square roots of negative numbers aren't real numbers.
In the context of our exercise, the function \(f(x) = \sqrt{x}\) only accepts non-negative values because you cannot take the square root of a negative number without delving into complex numbers.
This is why the domain of \(f \circ g\), where both \(f(x)\) and \(g(x)\) are \(\sqrt{x}\), is \(x \geq 0\).
In general, it's crucial to find the domain while manipulating functions to avoid errors and undefined expressions.

Graphing Utility

A graphing utility is a tool or software that helps visualize mathematical functions.
It allows you to input functions and view their graphs on a coordinate plane.
This can be incredibly valuable for understanding the behavior, shape, and intersections of functions.
When graphing \(f \circ g\) and \(g \circ f\), both of which are \(y = x^{1/4}\), a graphing utility can help illustrate the result in a simple and clear manner.
It helps confirm that these composite functions are indeed identical by showing they graph the same curve.
A graphing utility makes it easy to see connections and determine key features like intercepts and asymptotes.
It's particularly handy in validating our results from calculation, as it visually shows if our expected domain is accurate.

Square Roots

Understanding square roots is essential when dealing with functions like the ones in this exercise.
The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\).
So, \(\sqrt{x}\) is the principal square root, which is always non-negative.
Considering the exercise with \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{x}\), creating composite functions involves computing nested square roots, such as \(f(g(x)) = \sqrt{\sqrt{x}}\).
This expression simplifies to \(x^{1/4}\), meaning you're taking the fourth root of \(x\).
It's crucial to remember that square roots only accept non-negative inputs as valid within the real number system.

Equivalence of Functions

Equivalence of functions occurs when two different expressions lead to the same function.
This can be shown both algebraically and graphically.
In algebra, we find that \(f \circ g = x^{1/4}\) and \(g \circ f = x^{1/4}\) give the same result.
Thus, these composed functions are equivalent because, regardless of how they're approached mathematically, they simplify to the same formula.
When you use a graphing utility to plot these functions, you'll notice they overlay completely, verifying their equivalence visually.
This means that despite different paths or sequences in composition, their effect on any input \(x\) is the same.
Recognizing equivalence ensures that seemingly different operations result in consistent outputs, a cornerstone concept in mathematical analysis.